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How Back-to-Front Maths works:

Problem-based teaching is different to didactic teaching

Back-to-Front Maths begins with problem-solving, where students explore brand-new concepts and then use their findings to derive algorithms and formulae.  It works by creating light-bulb moments for students and enabling them to discover for themselves underlying mathematical principles, rather than providing explanation-and-practice pedagogy.

Here is a peer-reviewed academic paper from the Mathematics Education Research Group of Australasia conference in 2018 showing an effect size of 0.7 above the yearly expectations in PAT Maths over two years:  Kennedy – Intervention using challenging tasks and conceptual change

We have collected data from schools using Back-to-Front Maths either as student books or as website subscribers (2012-2015 for formal statistics – after that we have just continued to check that schools are moving towards green).  On average, our results show a statistically significant higher cohort growth for NAPLAN than those for similar schools.  To check out how much higher our results are, download these two documents:
Grade 3-5 data summary for Back to Front Maths
Grade 5-7 data summary for Back-to-Front Maths

For a more personal explanation, you might like to read What Back to Front Maths is all about and why I can’t market it


By beginning with problem-solving, teachers are able to diagnose student misconceptions and uncover holes in their intuitive understanding of mathematics.  These holes stop kids learning, as they try to hold ideas which oppose each other in their heads at the same time.

While didactic teaching occasionally uncovers misconceptions, problem-based teaching is designed to both uncover and lead students to self-correct these wrong ideas, creating deep and enduring understanding of mathematical principles.  Learning becomes far easier once students have an underlying structure on which to pin new concepts, but without this structure students tend to need constant drill and skill to remember procedures and formulae.

Read more about how to diagnose, confront and fix misconceptions here, including videos.

Teacher’s job

Instead of providing explanations, algorithms and formulae, teachers change focus to ask leading questions so that students develop mathematical principles for themselves.  These leading questions step students through a developmental sequence which forces them to abandon wrong ideas and to remember previously understood principles.  The sequence then builds on these principles to develop new mathematical thinking that can be trusted by the students as it is based on their own intuitive understanding. Accuracy and efficiency are built after conceptual understanding.


When students develop their own mathematical understanding it lasts.  The need for repetition and practice is greatly reduced, and the flexibility and reliability of the concepts is greatly increased.  Everything in mathematics is about patterns.  When students uncover these for themselves learning improves rapidly.  Some results that our schools have reported include:

  • Support students catching up to grade-level standard in under 12 months (see research article above)
  • Large NAPLAN improvements (see documents above)
  • Engaged students leading to excited and motivated teachers
  • Far fewer behaviour management issues – students who are engrossed in mathematics are too busy to remember to misbehave!

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Find out about Partnership Projects here, including sample reports to examine.

What online resources are available to subscribers?

Lessons bank:

This section enables users to download, print and use any activity from any grade with their students.  These lessons cover every requirement from the Australian Curriculum, including the proficiency strands, from Foundation to year 7.

Lesson plans for the Journal problems clearly explain how to implement problem-based teaching, with misconceptions, leading questions and suggestions for differentiation included for every problem.

Printable activities include Journal problems for introducing new concepts through problem-solving and diagnosing misconceptions, and Blast activities for using questioning to build the deep understanding necessary to develop mathematically.

Sample – Developmental sequence for fractions for year 3

Sample – Year 1 lesson on two-digit numbers

Sample – Fluency activity: Make ten dominoes

Free trial offer – download sample activities from each different grade here.

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Assessment in Back-to-Front Maths is based on the proficiency strands, just like in the achievement standards.  This section clearly and simply explains what is means to receive an A, B, C, D or E and what to look for to make these judgements.  This includes a range of criteria sheets, assessment schedules and explanations provided in both teacher and parent-friendly language.

Moderation tasks are also included for taking base-line, mid-year and end-year standardised measurements.  These tasks are designed with deliberately low-level content, but with very non-standard thinking, to more accurately pinpoint growth in problem-solving, reasoning and understanding.

Click here to download a sample moderating task for year 6

Click here to read about our Assessment and Moderation Training Course.

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Eight investigations for each year level from 3-7 have been included in this section.  Each investigation is designed to cover concepts from across at least three year levels.  These real-life tasks vary in length from a few hours to a whole term, and list journal problems and blast activities used to support the investigation.

Planning tools:

Alignment tables and term planners enable teachers to quickly and simply design units of work for their students using Microsoft WORD documents.  Detailed explanations of how to fit everything in, and how to provide differentiation for both support and extension students are also included in this section.


Free resources and activities to use with your students

This section contains both downloadable samples and links to the parts of the website that are not password-secured.

Download a free trial of sample lessons from each grade by entering your details in the form on the right hand side near the top of this page.

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